Dynamic Polar Coordinate Grid

• Dynamic polar coordinate grids are adaptive spatial discretizations that optimize coverage, resolution, and equivariance in non-Cartesian environments.
• They are widely applied in autonomous perception, fluid simulations, and astrophysical modeling to ensure uniform cell statistics and boundary adherence.
• Algorithmic and deep learning methods leverage these grids to boost simulation accuracy and efficiency, yielding significant performance gains in complex domains.

A dynamic polar coordinate grid is a spatial discretization scheme in which the underlying polar (or generalized polar-like) coordinate system is adaptively or algorithmically constructed to optimize coverage, resolution, or equivariance properties for non-Cartesian geometries, sensors, or physical systems. Such grids are essential in autonomous perception (e.g., radar/lidar-based occupancy maps), fluid and plasma simulations (e.g., toroidal MHD equilibria), and astrophysical simulations (e.g., binary black holes), enabling both efficient computation and precise resolution in physically meaningful domains.

1. Principles of Polar Grid Construction and Discretization

A canonical polar grid discretizes a 2D domain relative to an origin, parameterizing locations by radius $r$ and angle $\theta$. For example, in autonomous driving sensor grids (Wei et al., 2023, Zhang et al., 2020), each point $P$ with Cartesian coordinates $(x,y)$ is transformed via: $r = \sqrt{x^2 + y^2},\quad \theta = \arctan2(y, x)$ Discrete indices are computed as: $$i_r = \left\lfloor \frac{r}{\Delta r} \right\rfloor, \qquad i_\theta = \left\lfloor \frac{\theta + (A\,\Delta\theta)/2}{\Delta\theta} \right\rfloor$$ where $\Delta r$ and $\Delta\theta$ are bin sizes for radius and azimuth, respectively, and $A$ is the number of azimuth bins. The field-of-view defines $$\theta \in [-A\,\Delta\theta/2, +A\,\Delta\theta/2)$$ and $r \in [0, R_\mathrm{max}]$. By tuning $\Delta r$ and $\Delta\theta$, one aligns the grid's resolution with the sensing profile or spatial variation of the problem. This binning ensures that, for applications such as radar perception or lidar segmentation, cell occupancy statistics are approximately uniformized along the azimuth (Zhang et al., 2020).

In 3D or non-circular geometries, computational mappings generalize polar grids. For toroidal domains, the physical map $\Phi: (u,v,w)\to\mathbb{R}^3$ is defined as

$$\Phi(u,v,w) = R(u,v,w) [\cos v\, \mathbf{e}_x + \sin v\, \mathbf{e}_y] + Z(u,v,w)\, \mathbf{e}_z$$

for $(u,v)\in[0,2\pi]^2$, $w\in[0,1]$, capturing surfaces of revolution and boundary-conforming coordinates (Tecchiolli et al., 13 May 2024).

2. Dynamicity and Adaptivity in Polar Grids

The term "dynamic" refers to either:

• The application of time-dependent or state-dependent remappings (as in dynamic occupancy grids or time-varying simulations);
• The adaptive, algorithmic generation of coordinate grids that respond to environmental structure, sensor footprint, or computational efficiency constraints.

In radar-based dynamic occupancy mapping (Wei et al., 2023), the core grid discretization is fixed, but the occupancy estimates $M_\mathrm{polar}[i_\theta, i_r]$ and associated velocities $v_r$ are updated dynamically each time frame, resulting in a time-evolving grid-based scene representation. The fusion of static ISM output with Doppler-based velocity and Bayesian occupancy filtering realizes a fully dynamic, velocity-aware polar occupancy map.

For simulations, coordinate grids can not only be nonuniform, but also warped dynamically to track moving objects or focus resolution. In binary black hole simulations, "dynamic fisheye" mappings blend local grid densification near moving centers (e.g., black holes) and asymptote to canonical polar/spherical coordinates at large radii. The mapping is time-dependent, governed by kernel warping functions following the system's evolution (Zilhão et al., 2013).

In variational toroidal grid generation (Tecchiolli et al., 13 May 2024), the grid is dynamically constructed by action minimization to ensure boundary conformity and uniform Jacobian, thus avoiding coordinate singularity or grid tangling even for strongly shaped, nonaxisymmetric domains.

3. Algorithmic and Deep Learning Architectures Leveraging Polar Grids

Dynamic polar grids serve as domain-specific priors and enable the design of neural architectures that harness the geometry of the environment or sensor model.

In deep radar occupancy mapping, a ResNet-style encoder coupled to a spatial/channel dual attention decoder operates directly on the $(A\times R)$ polar grid (Wei et al., 2023). The input $Z_\mathrm{polar}$ (binary radar detection map) is mapped to an evidential belief grid $M_\mathrm{polar}$ via softmax and normalization, providing Dempster–Shafer masses for free, occupied, and unknown states.

For semantic lidar segmentation, PolarNet (Zhang et al., 2020) processes single-scan point clouds by quantizing them into $(N_r\times N_\theta)$ polar cells and using a pointwise MLP+maxpool per cell. The 2D "ring" feature map is processed by a U-Net with ring convolutions that wrap azimuthally—preserving polar grid periodicity and enabling efficient semantic decoding.

In medical segmentation (DDNet) (Liu et al., 2019), coordinate transforms and feature flows between Cartesian and polar domains are implemented via a differentiable polar transform layer (PTL). The PTL applies spatial transformer-style resampling, allowing the network to learn translation equivariant (Cartesian branch) and rotation equivariant (polar branch) features. Fusion is achieved through learned channel and spatial attention.

4. Variational and Physically-Constrained Dynamic Polar Grids

Dynamic polar-like grids for complex geometries are often constructed by minimizing a global action that controls the shape and orthogonality of coordinate surfaces.

The variational approach for toroidal domains (Tecchiolli et al., 13 May 2024) introduces an action functional

$$S[\Phi] = \int_0^1\!\int_0^{2\pi}\!\int_0^{2\pi} \left[ \tfrac{1}{2}f(\Phi,u,v,w)(\sqrt{g})^2 + \omega |\partial_w\Phi| \right] du\,dv\,dw$$

where $f$ is a weighting (e.g., $1/[w R^2]$), $g$ is the mapping Jacobian, and $\omega$ penalizes radial line curvature. Boundary conditions enforce conformity to physical domain boundaries and coordinate singularities (axis). The minimizer $\Phi$ is obtained by spectral coefficient descent until the discrete action and its gradient reach convergence with no Jacobian sign changes.

In general relativistic MHD (Zilhão et al., 2013), the dynamic fisheye mapping is constructed via smooth step- and box-car kernels (e.g., $\tau$, $\mathcal T$, based on hyperbolic tangent functions), allowing grid densification around moving loci. The map and its Jacobian are constructed analytically and updated at every time step to maintain focus near orbiting black holes.

5. Equivariance, Performance, and Application-Specific Benefits

Polar and dynamic polar grids afford important geometric properties:

• Rotation Equivariance: On a polar grid, input image rotations correspond to circular shifts along the azimuth axis, which is naturally handled by translation-equivariant network modules (convolutions, pooling) (Liu et al., 2019).
• Cell Load Balancing: The polar area of cells increases linearly with $r$, counteracting the $1/r$ fall-off in sensor point density (e.g., lidar), leading to more balanced cell occupancy statistics (Zhang et al., 2020).
• Boundary Adherence: Variational and warped grid schemes ensure that coordinates conform to challenging physical domains (toroidal or binary), avoiding coordinate singularities and promoting mesh regularity (Tecchiolli et al., 13 May 2024, Zilhão et al., 2013).

Performance improvements are application-dependent. In highway radar occupancy mapping, the deep polar-grid ISM achieves class-averaged mIoU of 68% (vs. geometric baseline 53%), occupied-cell precision of 39% (vs. 15%) and free-cell recall of 86% (vs. 60%) (Wei et al., 2023). For single-scan lidar segmentation, the polar U-Net representation boosts mIoU by 3–15 points over competing BEV and spherical approaches without significant computational penalties (Zhang et al., 2020).

6. Implementation Summary and Considerations

A summary of key construction and dynamic update procedures for polar grids is as follows:

Domain/Application	Grid Construction	Dynamicity/Adaptivity
Radar occupancy (Wei et al., 2023)	Discretize $(r, \theta)$, fixed $\Delta$	Dynamic occupancy (log-odds) + Doppler in $(A\times R)$ over time
LiDAR segmentation (Zhang et al., 2020)	Discretize $(r,\theta)$, fixed grid	Online, per-scan data mapping; preprocessing geometric adaptivity
Medical segmentation (Liu et al., 2019)	PTL on feature maps $(H\times W)$	Feature-level polar–Cartesian fusion via trainable PTL/fusion module
3D MHD/Plasma (Tecchiolli et al., 13 May 2024)	Variational minimization of action on $(u,v,w)$	Grid coefficients adjusted offline for each boundary shape
Black hole MHD (Zilhão et al., 2013)	Analytical time-dependent warped polar	Warping centers/parameters updated each timestep with system motion

Implementation caveats include performance tuning (balancing computational cost with accuracy), parameter selection for smoothness and Jacobian positivity, and the need for architecture-aware adjustment of grid size and channel/feature map handling.

7. Related Methods and Future Directions

Dynamic polar coordinate grids integrate ideas from spatial transformer networks, equivariant neural architectures, boundary-conforming mesh generation, and time-dependent warping approaches.

A plausible implication is that as sensor modalities and simulation domains grow in geometric complexity, further generalizations—incorporating non-uniform, data-driven, or learned polar (or more general non-orthogonal) grids—will be developed to accommodate multimodal, multi-centric, or non-Euclidean scenarios. The convergence of variational geometry-based grid generation (Tecchiolli et al., 13 May 2024), deep learning on adapted manifolds (Zhang et al., 2020, Liu et al., 2019), and real-time dynamic updating (Wei et al., 2023, Zilhão et al., 2013) exemplifies the broadening utilization of dynamic polar grids across computational science and engineering.